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### Is there a quick proof as to why the vector space of $\mathbb{R}$ over $\mathbb{Q}$ is infinite-dimensional?

It would seem that one way of proving this would be to show the existence of non-algebraic numbers. Is there a simpler way to show this?
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### Proving that $\left(\mathbb Q[\sqrt p_1,\dots,\sqrt p_n]:\mathbb Q\right)=2^n$ for distinct primes $p_i$.

I have read the following theorem: If $p_1,p_2,\dots,p_n$ are distinct prime numbers, then$$\left(\mathbb Q\left[\sqrt p_1,\dots,\sqrt p_n\right]:\mathbb Q\right)=2^n.$$ I have tried to prove a ...
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For $n\ge 3, x_{1},...,x_{n} \in \mathbf{Q}^{\ast}$ and $[\mathbf{Q}(\sqrt{x_{1}},...\sqrt{x_{n}}) : \mathbf{Q}] < 2^{n}$ how can we conclude that there are non empty $I \subset \{1,...,n\}$ with $... 2answers 97 views ### How to prove the set$\{\sqrt{n}:\textrm{$n$ is squarefree}\}$to be a linearly independent set? As the title goes, I am stuck on this problem. Prove that the set$\{\sqrt{n}:\textrm{$n$ is squarefree}\} =\{1,\sqrt{2}, \sqrt{3},\sqrt{5},\sqrt{6},\sqrt{7},\sqrt{10},\ldots\}$is a linearly ... 1answer 310 views ###$\sqrt{p_1}$is not in$Q[\sqrt{p_2},…,\sqrt{p_n}]$[duplicate] How to show$\sqrt{p_1}$is not in$Q[\sqrt{p_2},...,\sqrt{p_n}]$if$p_1,...,p_n$are distinct primes? Intuitively, this is pretty clear, but it makes me very uncomfortable to just believe. Any idea ... 1answer 69 views ### Splitting field of$(x^2-2)(x^6-20)$over$\mathbb{Q}$I have to determine the splitting field$K$of$f(x)=(x^2-2)(x^6-20)$over$\mathbb{Q}$. My attempt of solution:$K=\mathbb{Q}(\sqrt2, \sqrt[6]{20}, i\sqrt3)$;$d_1:=[\mathbb{Q}(\sqrt2, \sqrt[6]{20})(...
Consider $a_1,\ldots,a_n\in \mathbb Z$. i) Suppose $a_1,\ldots, a_n$ are pairwise relatively prime. I have to see by induction on n that $[\mathbb Q(\sqrt a_1,\ldots,\sqrt a_n):\mathbb Q]=2^n$ Once ...